The governing equations are solved in a three-dimensional rectangular domain using high-order finite difference methods. All spatial derivatives including convective and viscous terms in the streamwise and wall-normal directions are approximated with standard fourth-order compact difference schemes [1]}, except for the first derivative of non-linear terms in the \(x\) -direction. It is known that application of central compact difference discretizations to high Reynolds number flows typically leads to numerical instability. It is attributed to the accumulation of the aliasing errors resulting from discrete evaluation of the nonlinear convective terms in the \(x\) -direction, owing to the fact that central compact difference schemes are characterized by zero numerical dissipation error and therefore, are prone to aliasing error. A common practice to remove aliasing errors and hence enhancing the numerical stability is filtering the numerical solution at every time level. Instead, an upwind combined compact difference (CCD) scheme proposed in [2]} was employed. They demonstrated that their proposed upwind CCD scheme has non-zero dissipation error (numerical diffusion) restricted to the high wavenumber region, while exhibiting very good spectral resolution characteristic. The flow field is assumed to be periodic in the \(z\) -direction allowing the flow field to be expanded in Fourier cosine and sine series. A fourth-order accurate Runge–Kutta scheme is used to integrate the vorticity transport equations in time.
